Attosecond dynamics of multi-channel single photon ionization

Photoionization of atoms and molecules is one of the fastest processes in nature. The understanding of the ultrafast temporal dynamics of this process often requires the characterization of the different angular momentum channels over a broad energy range. Using a two-photon interferometry technique based on extreme ultraviolet and infrared ultrashort pulses, we measure the phase and amplitude of the individual angular momentum channels as a function of kinetic energy in the outer-shell photoionization of neon. This allows us to unravel the influence of channel interference as well as the effect of the short-range, Coulomb and centrifugal potentials, on the dynamics of the photoionization process.

Supplementary Note 1: Angular coefficients The angular coefficients in the general form C m l,l+1 are derived by calculating the angular transition matrix element between an initial state with angular momentum and a final state with angular momentum + 1: Supplementary Note 2: Continuum-continuum transition Applying and extending the results of previous work [1,2,3], we use the universal behavior of the cc-transitions to determine some of the terms in Eq. (3), thus reducing the number of unknown quantities. Supplementary Figure 1 (a,d) presents the ratios between the two-photon transition radial amplitudes, calculated as described in the methods section and defined as  Figure 1: Continuum-continuum transitions. Calculated amplitude ratios α ± λ (a,d) and continuum-continuum phases φ ± Lλ (b-c and e-f) for the absorption (a-c) and emission (d-f) processes. For the amplitude ratios, the curves correspond to different intermediate states (λ=1,2,3 in blue, red, green, respectively) and different atoms/initial states (square, He, = 0); (cross, Kr, = 2); (triangle, Ne, = 1); (circle, Ar, = 1). (b,e) refer to transitions with increasing angular momentum, L = λ + 1, while in (c,f), L = λ − 1 .
intermediate states (λ=1,2,3 in blue, red and green) and different atoms and initial states. The ratios show an universal behavior [3], independent of the atom. Only the orbital angular momentum of the intermediate state, λ, is of importance.
We present φ ± Lλ in Supplementary Figure 1 for increasing (b,e) and decreasing (c,f) angular momenta, in the absorption (b,c) and emission cases (e,f). The colors and symbols, corresponding to different intermediate angular momenta and atoms/initial states are indicated in the figure caption. These results show that the variation of the continuumcontinuum phase is universal, depending mainly on whether the IR photon is absorbed or emitted. For the absorption process, the cc-phase decreases as a function of kinetic energy and is positive, while for the emission, it increases and is negative. Note that the phases are not mirror image of each other, i.e. φ + The cc-phases depend on whether the angular momentum increases or decreases, especially at low kinetic energy, as observed by comparing (b) and (c), or (e) and (f). Finally, the cc-phases depend only slightly on the intermediate angular momentum (compare blue and red curves) and not at all on the atomic system (e.g. compare circle and triangle) in the range of energies studied here.  Table 1 for an initial = 1 state. We note that only the phase difference ∆Φ 2q = Φ 2q+1 − Φ 2q−1 , and not the individual high-order harmonic phases, plays a role in Eq. (3). The number of unknown quantities is therefore nine. While here the case of = 1 is shown, this is also true for higher initial angular momenta. Atomic phases ϕ01,2q∓1, ϕ21,2q∓1 φ ± 10 , φ ± 12 , φ ± 32 , from Supp. Fig. 1(b,c,e,f)

(a,d)
Harmonic phases We determine these nine unknown quantities using a global fit to our experimental measurements. In general, multiphoton electron angular distributions can be written as an expansion in Legendre polynomials [2,4,5,6]. For a two-photon transition, without parity mixing, the expansion needs only three polynomials, P 0 (x) = 1, P 2 (x) = (3x 2 − 1)/2, and P 4 (x) = (35x 4 − 30x 2 + 3)/8, reading as The theoretical expressions for the coefficients h i (τ ), i = 0, 2, 4, can be obtained by expanding Eq. (3) and replacing the products of spherical harmonics by Legendre polynomials, leading to the following equations: The coefficients h i (τ ) are extracted from the data by a fit of equation S3 to the background subtracted and normalized photoelectron angular distributions (PADs) for each delay. Supplementary Figure 2 shows the variation of h i (τ ) extracted from the experimental data for each delay (black points). As is clear from Eqs. (S4-S6) and from Supplementary Figure 2, each h i (τ ) oscillates with the delay τ at the frequency 2ω and is therefore fully determined by three quantities: mean value, amplitude and phase. Thus, a total of nine parameters describe the angle and delay dependence of the sideband signal I SB (θ, τ ). This implies that the nine unknown quantities in table 1 can be determined through a global fit of the three analytical expressions of h i (τ ), to the experimentally measured coefficients in Supplementary Figure 2. This simultaneous fit to the derived system of equations is based on a Levenberg-Marquardt algorithm using a least square regression and extensively tested for its convergence using theoretical data. The result of such a global fit is shown by the red lines in Supplementary Figure 2.
The nine unknown quantities in table 1 are not completely independent, since the onephoton ionization phases only appear as differences in Eqs. (S4-S6). We therefore lock ϕ 01,15 for the first harmonic order (15) and determine the other three phases involved in this measurement (ϕ 21,15 , ϕ 01,17 , ϕ 21,17 ). The global fit is repeated for the next sideband (18), which involves a contribution from the same harmonic order (17) as sideband 16. Using the previously determined phases for harmonic 17, we then retrieve the one-photon phases for harmonic 19. This process is iteratively repeated over the entire energy range allowing us to map out the energy dependence of the one-photon phases.